Consider any Lie group $G$ and denote by $\mathfrak{g}^*$ the dual space of the Lie algebra of $G$. And consider the usual manifold structure on $\mathfrak{g}^*$. Then, I've heard somewhere that the vector space of differentiable functions on $\mathfrak{g}^*$ is generated by the linear functions.
In what sense could this be true? How we can prove it?